Convex cones

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Nonorthogonal projection on extreme directons of convex cone

pseudo coordinates

Let LaTeX: \mathcal{K} be a full-dimensional closed pointed convex cone in finite-dimensional Euclidean space LaTeX: \mathbb{R}^n.

For any vector LaTeX: \,v\, and a point LaTeX: \,x\!\in\!\mathcal{K}, define LaTeX: \,d_v(x)\, to be the largest number LaTeX: \,t^\star such that LaTeX: \,x-t^{}v\!\in\!\mathcal{K}\,.

Suppose LaTeX: \,x\, and LaTeX: \,y\, are points in LaTeX: \,\mathcal{K}\,.

Further, suppose that LaTeX: \,d_v(x)\!=_{\!}d_v(y)\, for every extreme direction LaTeX: \,v\, of LaTeX: \,\mathcal{K}\,.

Then LaTeX: \,x\, must be equal to LaTeX: \,y\,.

proof

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